On the definition of fluctuating temperature

TL;DR

This paper derives the Maxwell distribution from the F-distribution, clarifies the interpretation of temperature estimators and Lagrange multipliers in statistical mechanics, and contrasts frequency and Bayesian views on fluctuating temperature.

Contribution

It establishes the conditions under which the Lagrange multiplier can be interpreted as inverse temperature and highlights the limitations of nonadditive entropy approaches.

Findings

The Maxwell distribution arises from the F-distribution in the limit of infinite degrees of freedom.

The temperature estimator is consistent, unbiased, and efficient, matching the heat reservoir temperature asymptotically.

No generalization of chi-squared distributions allows interpreting the Lagrange multiplier as inverse temperature.

Abstract

The Maxwell distribution is derived from the $F$-distribution in the limit where one of the degrees of freedom of the $\chi^2$ variates tends to infinity. The estimator of the temperature is consistent, and, hence coincides with the temperature of the heat reservoir in the asymptotic limit; it is also unbiased and efficient. Consequently, there is a contradiction between indentifying the Lagrange multiplier in the variational formalism that Tsallis and co-workers use to maximize his nonadditive entropy with respect to escort expectation values in order to derive the Student $t$- and $r$-distributions and the physical meaning of these variables. Only in the asymptotic limit when these distributions become the $\chi^2$-distributions of MBG statistics can the Lagrange multiplier be interpreted as the inverse temperature. Hence, there is no generalization of the $\chi^2$-distributions that can be made which involves interpreting the Lagrange multiplier as the inverse temperature. The frequency interpretation of the fluctuating temperature is contrasted with the Bayesian approach that treats a parameter to be estimated as a random variable which is equipped with a probability distribution.